John Stillwell - Naive Lie Theory.pdf - Index of

8.1 Open and closed sets in Euclidean space 163
The relative topology
Many spaces S other than R k have a notion of distance, so the definition
of open and closed sets in terms of ε-balls may be carried over directly. In
particular, if S is a subset of some R k we have:
• The ε-balls of S , N ε (P)={Q ∈ S : |P − Q| < ε}, are the intersections
of S with ε-balls of R k .
• So the open subsets of S are the intersections of S with the open
subsets of R k .
• So the closed subsets of S are the intersections of S with the closed
subsets of R k .
The topology resulting from this definition of open set is called the relative
topology on S . It is important at a few places in this chapter, notably for
the definition of a matrix Lie group in the next section.
Notice that S is automatically a closed set in the relative topology,
since it is the intersection of S with a closed subset of R k , namely R k
itself. This does not imply that S contains all its limit points; indeed, this
happens only if S is a closed subset of R k .
Exercises
Open sets and closed sets are common in mathematics. For example, an open
interval (a,b)={x ∈ R : a < x < b} is an open subset of R and a closed interval
[a,b]={x ∈ R : a ≤ x ≤ b} is closed.
8.1.1 Show that a half-open interval [a,b)={x : a ≤ x < b} is neither open nor
closed.
8.1.2 With the help of Exercise 8.1.1, or otherwise, give an example of an infinite
union of closed sets that is not closed.
8.1.3 Give an example of an infinite intersection of open sets that is not open.
Since a random subset T of a space S may not be closed we sometimes
find it convenient to introduce a closure operation that takes the intersection of all
closed sets F ⊇ T :
closure(T )=∩{F ⊆ S : F is closed and F ⊇ T }.
8.1.4 Explain why closure(T ) is a closed set containing T .
8.1.5 Explain why it is reasonable to call closure(T ) the “smallest” closed set
containing T .
8.1.6 Show that closure(T )=T ∪{limit points of T } when T ⊆ R k .